Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 10.

1. Problem 1.

Let G = {g_{ij}}_{i,j=1,2} and B = {b_{ij}}_{i,j=1,2} be the matrices of two quadratic forms G and B on the plane, where G is positive definite, and g_{12} = g_{21}, b_{12} = b_{21}. Consider a quadratic (in λ) equation det(B − λG) = 0. The roots λ_{1} and λ_{2} of this equation are called the eigenvalues of the pair of quadratic forms G and B.

a) Show that if λ_{1} and λ_{2} are the eigenvalues of G and B, then two systems of two equations

\{ \begin{matrix}(b_{11} − λ_{i}g_{11})ξ_{i}^{1} + (b_{12} − λ_{i}g_{12})ξ_{i}^{2} = 0 \\ (b_{12} − λ_{i}g_{12})ξ_{i}^{1} + (b_{22} − λ_{i}g_{22})ξ_{i}^{2} = 0, \end{matrix}

i = 1, 2, with two unknowns f_{1} = (ξ_{1}^{1}, ξ_{1}^{2}) and f_{2} = (ξ_{2}^{1}, ξ_{2}^{2}) has a non-trivial solution.

The directions of the vectors f_{1} and f_{2} are called the main directions of the pair of quadratic forms; f_{1} corresponds to λ_{1} and f_{2} corresponds to λ_{2}.

In b) - e) you will show that If the eigenvalues of the pair of quadratic forms are different, then the main directions are orthogonal. In other words, you will prove that

f_{1} · f_{2} = ∑_{i=1}^{2} ∑_{j=1}^{2} g_{ij}ξ_{1}^{i}ξ_{2}^{j} = 0,  (1)

provided λ_{1} ≠ λ_{2}. Here f_{1} = ξ_{1}^{1}e_{1} + ξ_{1}^{2}e_{2}, f_{2} = ξ_{2}^{1}e_{1} + ξ_{2}^{2}e_{2}, and e_{1} and e_{2} are the basic vectors in the plane such that e_{i} · e_{j} = g_{ij}, 1 ≤ i, j ≤ 2.

b) Prove that there exists a basis d_{1}, d_{2} such that d_{i} · d_{j} = δ_{ij}, where δ_{ij} = 0 if i ≠ j and δ_{ij} = 1 if i = j.

c) Show that in the new basis d_{1}, d_{2} for the matrices Ḡ = I and B̄ (of G and B) one

has G = A^{T} ḠA and B = A^{T} B̄A, where A = {a_{ij}}_{i,j=1,2} is such that e_{i} = ∑_{j=1}^{2} a_{ij}d_{j},

i = 1, 2.

d) Use c) to show that

det(B − λG) = (det A)^{2} det(B̄ − λI), det A ≠ 0,

and deduce that the conditions

det(B − λG) = 0 det(B̄ − λI) = 0

are equivalent.

e) Prove that the quadratic form B̄ can be brought to a form corresponding to a diagonal matrix, say W. Prove that the eigenvectors of W are orthogonal in the usual Euclidean sense. Conclude that (1) holds.

2. Problem 2. Let S_{p} : T_{p}(S) → T_{p}(S) be the shape operator defined by S_{p}(v) = −D_{v}n(p).

a) Prove that S_{p} is linear.

b) Show that the matrix representing S_{p} with respect to the basis {r_{u}, r_{v}} is

II_{p}I_{p}^{−1} = ( \begin{matrix}L & M \\ M & N \end{matrix} ) ( \begin{matrix}E & F \\ F & G \end{matrix} )^{−1} .

Hint: Write S_{p}(r_{u}) = ar_{u} + br_{v} and S_{p}(r_{v}) = cr_{u} + dr_{v}, and use the definition of L, M and N to get a system of linear equations for a, b, c and d.

c) Deduce that

K = \frac{LN − M^{2}}{EG − F^{2}} .

3. Problem 3. Compute the second fundamental forms II_{p} of the following parametrized surfaces. Then calculate the matrix of the shape operator, determine H and K.

a) the cylinder

r(u, v) = (a cos u, a sin u, v), a > 0.

b) the torus

r(u, v) = ((a + r cos u) cos v, (a + r cos u) sin v, r sin u), 0 < r < a.

c) the helicoid

r(u, v) = (u cos v, u sin v, bv), b > 0.

d) the catenoid

r(u, v) = a(cosh u cos v, cosh u sin v, u).